Conditional independence and probabilistic influence diagrams
Richard E. Barlow, Carlos Alberto de Bragança Pereira · Lecture notes-monograph series · 1990
A graphical approach to conditional independence is discussed.Some well known results concerning conditional independence are proved using simple influence diagram arguments.This material is, in part, from a book in progress tentatively titled Applied Bayesian Statistics, by the present authors. Introduction.Influence diagrams with decision nodes were invented in 1976 by Miller et al. [cf.Howard and Matheson (1984)].Shachter (1986) further developed methods for analyzing influence diagrams.S. Wright (1934) used diagrams to aid in understanding his "method of path coefficients."Although his diagrams pictorially resemble Gaussian influence diagrams [cf.Shachter and Kenley (1988)], they are not based on the Bayesian paradigm.They are not in any sense influence diagrams.I.J.Good (1961) invented "causal nets" that resemble influence diagrams.He used them to illustrate his ideas of causality and conditional independence.In this respect they are similar to influence diagrams.However he did not develop a comparable methodology for analyzing the diagrams.His diagrams are not influence diagrams as we define them below.Influence diagrams are useful for modeling statistical problems.Construction of the diagram is helpful in understanding the problem and communicating the interdependencies to others.In the process of constructing the influence diagram, a representation of the joint distribution of random quantities related to the problem of interest is developed.Usually one does not start with the joint distribution but uses the influence diagram model to determine a useful representation of the joint distribution.In the case of decision influence diagrams, the diagram can be used to help solve the decision problem(s) of interest.Examples of the use of influence diagrams can be found in Barlow and Zhang (1987) and Lauritzen and SpiegelhaJter (1988).